Albert Teen YOU ARE LEARNING:  Special Sequences: Geometric Sequences  # Special Sequences: Geometric Sequences ### Special Sequences: Geometric Sequences

Geometric sequences multiply each term by the same factor to produce the next term. We can use this idea to identify the nth term in a geometric sequence.

A geometric sequence is a sequence where each successive term is the result of multiplying the previous term by the same value, each time.

1

Let's look at an example

In this sequence, each term is $3 \times$ the previous term. 2

What would the next term in this sequence be?  3

The next term would be 243

Since each term is $3 \times$ the previous term, $3 \times 81=243$. 4

Each of these terms are powers of 3

$3^1=3$, $3^2=9$, $3^3=27$ and $3^4=81$. We can use this information to find the nth term for this sequence. 5

The nth term would be $3^n$

Remember, $n$ is the position in the sequence. Therefore, each number is a successive power of 3. What is the next term in this sequence?

$2,4,8,16,32$ In a geometric sequence, the ratio between each pair of successive terms is the same.

Let's look at another geometric sequence: $100,50,25,12.5$

1

Look at the values to find the difference

Each value is half of the previous value in the sequence. Therefore, it is a geometric sequence because each value is multiplied by the same number to produce the next value in the sequence.

2

What would be the next value in the sequence? 3

So how would we find the nth term?

Each value is multiplied by $\dfrac{1}{2}$ in order to produce the next value in the sequence.

4

The nth term phrase starts with 100 (the first term)

To produce the second term in the sequence, we multiply $100 \times 0.5$. To get to the third term in the sequence (25), we multiply by $0.5$, then multiply by $0.5$ again. Multiplying by $0.5 \times 0.5$ is the same as $0.5^2$.

5

The nth term is $a_n=100 \times 0.5^{n-1}$

To make the fourth term (12.5), we multiply 100 by $0.5 \times 0.5 \times 0.5$, which is the same as $0.5^3$. Therefore, we can see the pattern is to multiply the first value in the sequence by $0.5^{n-1}$, so the nth term is $100\times0.5^{n-1}$.

6

The nth term is $100 \times 0.5^{n-1}$. What is the 8th term to 3 dp? 